# What precisely does the notation $\frac {\partial h} {\partial z}$ mean in this context?

I am given the following problem.

Let $$Ω\subset \mathbb{C}$$ and let $$h \in C^1(Ω)$$ be such that $$\frac {\partial h} {\partial z}=0$$. Show that $$h(z)=\overline {f(z)}$$ for some $$f$$ analytic in $$Ω$$.

The $$\frac {\partial h} {\partial z}$$ notation is mysterious to me. But, after looking at this, I have a guess as to what it means.

Write $$h(x+iy)=u(x,y)+iv(x,y).$$ Then $$\frac {\partial h} {\partial z}=\frac {[u_x+iv_x]-i[u_y+iv_y]} 2=\frac {u_x+v_y+iv_x-iu_y} 2$$. Is this interpretation correct?

• It's just the ordinary derivative $\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - \frac{\partial}{\partial y}\right)$. – anomaly Jan 24 at 23:16
• @anomaly you can write that as an answer. It doesn't need to contain anything more – punctured dusk Apr 10 at 18:01